Unirationality and $R$-equivalence for conic bundles over quasi-finite fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929608791687168 |
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| author | Boughattas, Elyes |
| author_facet | Boughattas, Elyes |
| contents | Yanchevskiĭ had asked whether conic bundle surfaces over $\mathbf{P}^1_k$ are unirational when $k$ is a finite field. We give a partial answer to his question by showing that for quasi-finite fields $k$ (e.g. finite fields) a regular conic bundle $X$ over $\mathbf{P}^1_k$ is unirational if all non-split fibres lie over rational points. For large finite fields $k$, this beats a previous result of Mestre. Under the same assumption, we also prove that all rational points of $X$ are $R$-equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19686 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unirationality and $R$-equivalence for conic bundles over quasi-finite fields Boughattas, Elyes Algebraic Geometry Number Theory 14M20, 14E08 (Primary) 14D10, 14G12, 14G05, 14D06, 14H30 (Secondary) Yanchevskiĭ had asked whether conic bundle surfaces over $\mathbf{P}^1_k$ are unirational when $k$ is a finite field. We give a partial answer to his question by showing that for quasi-finite fields $k$ (e.g. finite fields) a regular conic bundle $X$ over $\mathbf{P}^1_k$ is unirational if all non-split fibres lie over rational points. For large finite fields $k$, this beats a previous result of Mestre. Under the same assumption, we also prove that all rational points of $X$ are $R$-equivalent. |
| title | Unirationality and $R$-equivalence for conic bundles over quasi-finite fields |
| topic | Algebraic Geometry Number Theory 14M20, 14E08 (Primary) 14D10, 14G12, 14G05, 14D06, 14H30 (Secondary) |
| url | https://arxiv.org/abs/2410.19686 |